On real-analytic Levi-flat hypersurfaces and holomorphic webs

TL;DR

This paper studies holomorphic webs tangent to real-analytic Levi-flat hypersurfaces on compact complex surfaces, proving the existence of meromorphic first integrals and the algebraic extension of Levi foliations under certain conditions.

Contribution

It establishes conditions under which holomorphic webs admit meromorphic first integrals and extends Levi foliations to algebraic webs on projective space.

Findings

Holomorphic webs tangent to Levi-flat hypersurfaces admit multiple-valued meromorphic first integrals.

Levi foliations induced by real-analytic curves extend to algebraic webs on projective space.

Results apply to compact complex surfaces with Levi-flat hypersurfaces.

Abstract

We investigate holomorphic webs tangent to real-analytic Levi-flat hypersurfaces on compact complex surfaces. Under certain conditions, we prove that a holomorphic web tangent to a real-analytic Levi-flat hypersurface admits a multiple-valued meromorphic first integral. We also prove that the Levi foliation of a Levi-flat hypersurface induced by an irreducible real-analytic curve in the Grassmannian $G(n+1,n)$ extends to an algebraic web on the complex projective space.